Flatpack Sphere

A simple sphere with 5 planes in 2 directions... Hand made in illustrator.

Can anyone recommend me a tool (commercial or free, but I prefer free :-) that will allow me to make planar cuts through organic shapes to be able to build them via laser cutter? For instance plug-ins for Blender or likewise. And pardon me if it already exists in Blender, I've only just started kicking around in it...

Just thinking, there has to be an easier way than keeping track of 3 planes in my head and hand drawing everything in Illustrator?